Estimates in quadratic formulas
type: Item
instance of: scholarly article
title: Estimates in quadratic formulas
zbMATH Open document ID: 0822.65022
DOI: 10.1007/BF02142693
author: Gene H. Golub
author: Zdeněk Strakoš
published in: Numerical Algorithms
publication date: 1995-03-12T00:00:00Z
review text: It is shown that the problem of estimating quadratic forms can be considered as a problem of computing Gauss and Gauss-Radau quadratures and this problem can be solved using the Lanczos method. Also, the effect of rounding errors to the Gauss quadrature calculations for the distribution function with finite points of increase is analyzed and it is proved that the total error of the Gauss quadrature computed via the conjugate gradient method in finite precision arithmetic can be described as truncation error of the Gauss quadrature for a different distribution function. As a consequence, it is shown that for the particular case \(f(\lambda)= 1/\lambda\), the total accuracy of the Gauss quadrature, computed by the finite precision conjugate gradient process, is determinated by the energy norm of the error of the conjugate gradient process.
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 15A63
Mathematics Subject Classification ID: 41A55
zbMATH DE Number: 720553
zbMATH Keywords: quadratic forms
zbMATH Keywords: Gauss-Radau quadratures
zbMATH Keywords: Lanczos method
zbMATH Keywords: Gauss quadrature
zbMATH Keywords: conjugate gradient method
reviewed by: Florica Luban
MaRDI profile type: Publication
cites work: Bounds for the error of linear systems of equations using the theory of moments
cites work: Q3323133
cites work: On Adaptive Weighted Polynomial Preconditioning for Hermitian Positive Definite Matrices
cites work: Construction of Gauss-Christoffel Quadrature Formulas
cites work: On Generating Orthogonal Polynomials
cites work: Quadratically constrained least squares and quadratic problems
cites work: Q4288554
cites work: Calculation of Gauss Quadrature Rules
cites work: Some Modified Matrix Eigenvalue Problems
cites work: Bounds for matrix moments
cites work: Estimates of Eigenvalues for Iterative Methods
cites work: Behavior of slightly perturbed Lanczos and conjugate-gradient recurrences
cites work: Predicting the Behavior of Finite Precision Lanczos and Conjugate Gradient Computations
cites work: Q3909906
cites work: Methods of conjugate gradients for solving linear systems
cites work: The Pauli algebra approach to special relativity
cites work: On the calculation of Jacobi matrices
cites work: Geometry of moment spaces
cites work: On the convergence rate of the conjugate gradients in presence of rounding errors
cites work: Computational Variants of the Lanczos Method for the Eigenproblem
cites work: Error Analysis of the Lanczos Algorithm for Tridiagonalizing a Symmetric Matrix
cites work: Accuracy and effectiveness of the Lanczos algorithm for the symmetric eigenproblem
cites work: Variational bounds on the entries of the inverse of a matrix
cites work: Relaxationsmethoden bester Strategie zur Lösung linearer Gleichungssysteme
cites work: The Lanczos Algorithm With Partial Reorthogonalization
cites work: Analysis of the symmetric Lanczos algorithm with reorthogonalization methods
cites work: On the real convergence rate of the conjugate gradient method
cites work: Q3290777
Recommended article: On quadrature formulas
Recommended article: ON QUADRATURE FORMULAS
Recommended article: Q3364298
Recommended article: Q3738462
Recommended article: Quadrature formulae and polynomial inequalities
Recommended article: Q5462265
Recommended article: About some quadrature formulas
Recommended article: Estimates for the remainders of certain quadrature formulas
Recommended article: Estimating a general function of a quadratic function
description: scientific article; zbMATH DE number 720553